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Theorem onexlimgt 41763
Description: For any ordinal, there is always a larger limit ordinal. (Contributed by RP, 1-Feb-2025.)
Assertion
Ref Expression
onexlimgt (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥𝐴𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem onexlimgt
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 omelon 9623 . . . 4 ω ∈ On
2 onun2 6461 . . . 4 ((𝐴 ∈ On ∧ ω ∈ On) → (𝐴 ∪ ω) ∈ On)
31, 2mpan2 689 . . 3 (𝐴 ∈ On → (𝐴 ∪ ω) ∈ On)
4 onexomgt 41761 . . 3 ((𝐴 ∪ ω) ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
53, 4syl 17 . 2 (𝐴 ∈ On → ∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
6 simp2 1137 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝑎 ∈ On)
7 omcl 8518 . . . . 5 ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o 𝑎) ∈ On)
81, 6, 7sylancr 587 . . . 4 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (ω ·o 𝑎) ∈ On)
9 noel 4326 . . . . . . . . . 10 ¬ (𝐴 ∪ ω) ∈ ∅
10 oveq2 7401 . . . . . . . . . . . . . 14 (𝑎 = ∅ → (ω ·o 𝑎) = (ω ·o ∅))
11 om0 8499 . . . . . . . . . . . . . . 15 (ω ∈ On → (ω ·o ∅) = ∅)
121, 11ax-mp 5 . . . . . . . . . . . . . 14 (ω ·o ∅) = ∅
1310, 12eqtrdi 2787 . . . . . . . . . . . . 13 (𝑎 = ∅ → (ω ·o 𝑎) = ∅)
1413eleq2d 2818 . . . . . . . . . . . 12 (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ (𝐴 ∪ ω) ∈ ∅))
1514notbid 317 . . . . . . . . . . 11 (𝑎 = ∅ → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
1615adantl 482 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → (¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎) ↔ ¬ (𝐴 ∪ ω) ∈ ∅))
179, 16mpbiri 257 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ¬ (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
1817pm2.21d 121 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝑎 ∈ On) ∧ 𝑎 = ∅) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎)))
1918ex 413 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑎 ∈ On) → (𝑎 = ∅ → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → Lim (ω ·o 𝑎))))
2019com23 86 . . . . . 6 ((𝐴 ∈ On ∧ 𝑎 ∈ On) → ((𝐴 ∪ ω) ∈ (ω ·o 𝑎) → (𝑎 = ∅ → Lim (ω ·o 𝑎))))
21203impia 1117 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ → Lim (ω ·o 𝑎)))
22 limom 7854 . . . . . . . . 9 Lim ω
231, 22pm3.2i 471 . . . . . . . 8 (ω ∈ On ∧ Lim ω)
246, 23jctir 521 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)))
25 omlimcl2 41762 . . . . . . 7 (((𝑎 ∈ On ∧ (ω ∈ On ∧ Lim ω)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
2624, 25sylan 580 . . . . . 6 (((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) ∧ ∅ ∈ 𝑎) → Lim (ω ·o 𝑎))
2726ex 413 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (∅ ∈ 𝑎 → Lim (ω ·o 𝑎)))
28 on0eqel 6477 . . . . . 6 (𝑎 ∈ On → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
296, 28syl 17 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝑎 = ∅ ∨ ∅ ∈ 𝑎))
3021, 27, 29mpjaod 858 . . . 4 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → Lim (ω ·o 𝑎))
31 simp1 1136 . . . . . 6 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ On)
3231, 8jca 512 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On))
33 simp3 1138 . . . . . 6 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ∪ ω) ∈ (ω ·o 𝑎))
34 ssun1 4168 . . . . . 6 𝐴 ⊆ (𝐴 ∪ ω)
3533, 34jctil 520 . . . . 5 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → (𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)))
36 ontr2 6400 . . . . 5 ((𝐴 ∈ On ∧ (ω ·o 𝑎) ∈ On) → ((𝐴 ⊆ (𝐴 ∪ ω) ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎)))
3732, 35, 36sylc 65 . . . 4 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → 𝐴 ∈ (ω ·o 𝑎))
38 limeq 6365 . . . . . 6 (𝑥 = (ω ·o 𝑎) → (Lim 𝑥 ↔ Lim (ω ·o 𝑎)))
39 eleq2 2821 . . . . . 6 (𝑥 = (ω ·o 𝑎) → (𝐴𝑥𝐴 ∈ (ω ·o 𝑎)))
4038, 39anbi12d 631 . . . . 5 (𝑥 = (ω ·o 𝑎) → ((Lim 𝑥𝐴𝑥) ↔ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))))
4140rspcev 3609 . . . 4 (((ω ·o 𝑎) ∈ On ∧ (Lim (ω ·o 𝑎) ∧ 𝐴 ∈ (ω ·o 𝑎))) → ∃𝑥 ∈ On (Lim 𝑥𝐴𝑥))
428, 30, 37, 41syl12anc 835 . . 3 ((𝐴 ∈ On ∧ 𝑎 ∈ On ∧ (𝐴 ∪ ω) ∈ (ω ·o 𝑎)) → ∃𝑥 ∈ On (Lim 𝑥𝐴𝑥))
4342rexlimdv3a 3158 . 2 (𝐴 ∈ On → (∃𝑎 ∈ On (𝐴 ∪ ω) ∈ (ω ·o 𝑎) → ∃𝑥 ∈ On (Lim 𝑥𝐴𝑥)))
445, 43mpd 15 1 (𝐴 ∈ On → ∃𝑥 ∈ On (Lim 𝑥𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wrex 3069  cun 3942  wss 3944  c0 4318  Oncon0 6353  Lim wlim 6354  (class class class)co 7393  ωcom 7838   ·o comu 8446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708  ax-inf2 9618
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-oadd 8452  df-omul 8453
This theorem is referenced by: (None)
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